Let $X$ be a compact manifold which admits an embedding in the projective space and let $\pi: Y \to X$ a projective bundle on it. I'm trying to prove that also $Y$ is algebraic (embeddable in projective space) but I do not really have idea how to do it(without using overkill method such as GAGA).
I tried to pullback the Kahler metric on $Y$ in a concrete way but did not really make up with anything.